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1, 1, 4, 49, 1764, 184041, 55294096, 47675849104, 117727187246656, 831443906113411600, 16779127803917965290000, 966945347924006310543140625, 159045186822042363450404006250000, 74638947576233124529271587010756250000, 99910846988474589225795290311922220324000000
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OFFSET
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0,3
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COMMENTS
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Expansion of generating function A_{QT}^(1)(4n).
a(n) is the number of cyclically symmetric and self-complementary plane partitions in a (2n)-cube. - Peter J. Taylor, Jun 17 2015
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REFERENCES
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D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.16), p. 199.
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LINKS
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FORMULA
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a(n) = 2^n * det U(n), where U(n) is the n X n matrix with entry (i, j) equal to binomial(i+j, 2*i-j)/2 + binomial(i+j, 2*i-j-1). [Ciucu]
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MATHEMATICA
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f[n_]:=Product[((3 k + 1)!/(n + k)!)^2, {k, 0, n-1}]; Table[f[n], {n, 0, 15}] (* Vincenzo Librandi, Jun 18 2015 *)
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PROG
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(PARI) a(n) = 2^n*matdet(matrix(n, n, i, j, i--; j--; binomial(i+j, 2*i-j)/2+binomial(i+j, 2*i-j-1))); \\ Michel Marcus, Jun 18 2015
(Magma) [n eq 0 select 1 else &*[(Factorial(3*k+1)/Factorial(n+k))^2: k in [0..n-1]]: n in [0..15]]; // Bruno Berselli, Jun 23 2015
(Python)
from math import prod, factorial
def A049503(n): return (prod(factorial(3*k+1) for k in range(n))//prod(factorial(n+k) for k in range(n)))**2 # Chai Wah Wu, Feb 02 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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